On Twitter right now I am soliciting thoughts about calculus courses, the topics we cover in them, and the ways in which we cover them. It’s turning out that 140 characters isn’t enough space to frame my question properly, so I’m making this short post to do just that. Here it is:

Suppose that you teach a calculus course that is designed for a general audience (i.e. not just engineers, not just non-engineers, etc.). Normally the course would be structured as a 4-credit hour course, meaning four 50-minute class meetings per week for 14 weeks. Now, suppose that the decision has been made to cut this to TWO credit hours, or 100 minutes of contact time per week for 14 weeks.

Questions: What topics do you remove from the course? What topics do you keep in the course at all costs? And of those topics you keep, do you teach them the same way or differently? If differently, then how would you do it? Finally, would there be anything NEW you’d introduce in the course that would be pertinent for a 2-hour course that wouldn’t show up in a 4-hour version of that course?

Keep Twittering your comments to me at @RobertTalbert, or comment below. I’ll sum them up later.

UPDATE: I also meant to say, feel free to play with the assumptions I am making here. For example, if it’s impossible to think of a 2-hour calculus course, change that to a 3-credit course and see if you can come up with anything.

7 responses to “A calculus thought experiment”

It would help to know whether this were just differntial or also include integral, which is pretty ridiculous, but I will assume just the former.

In this case I think that I would take the approach (advocated by Jerry King in his new book “Mathematics in 10 Lessons”)of teaching fundamentals that allows one to derive lots of other results. It is sort of a pure mathemetics approach but can be framed in such a way that applications naturally fall out of it, and such practice can be optionally attained outside of class with other materials. I personally like the text is “How to Ace Calculus.”

Anyway, I would approach things historically, that is, why the need for instantaneous rates of change, Newton – Leibniz controversy, Weierstrauss and Cauchy’s need to make things rigorous, Abraham Robinson’s hyperreals to the rescue, etc. I would include info on evaluating limits of functions and the nature of differentials and how the 2 methods were at odds for a while (remembering Berkeley’s criticisms – “the ghosts of departed quantities…” Controversy always makes things more interesting and relevant, especially if you can infuse things with philosophy and even divine ramifications (infinity and infinitesimals, incompleteness, Russell paradoxes, etc.).

Now all of this can be taught – I have done it at the high school level – in a non-overwhelming way. It just takes a little time to cover epsilon – delta, but that is key to the philospohy. Then, with overwhelming appreciation, I taught the rest using differentials.

Finally, in such a 2 hour course, you can’t get hung up going through tricky problems which take up a lot of class time, don’t really teach anything other than the application of cleverness. Really a section on using integral tables and a day spent on a software package should complete the course.

For the record, assume that the topics in the 4-hour version of the course correspond to the typical outline of an Advanced Placement Calculus AB course — the usual stuff from differential calculus plus the definition of the definite integral, the Fundamental Theorem and applications to calculating total change, calculating integrals with antiderivatives up through u-substitution, and basic applications of integration like area between curves. (No integration by parts, trig integration, volumes of revolution, etc.)

I want to say that I really like this post and I can’t wait for what people come up with. I recently had the exact same thought – of overhauling the calc class I teach (http://samjshah.com/2009/04/23/my-favorite-book-title/). I think I’m going to do this exercise this summer, when I can clear my mind and be systematic. If/when I do it, I’ll post my results too.

Upon further thinking on the topic of curricula, I do know if the math educator can ultimately prevail. There are too many debates raging. Most non-math majors and HS students are not going to embrace anything that teaches fundamentals, principles, concepts or whatever else you call them. They want rote mechanistic instruction and formulas. Recall the introduction of new math. What a disaster. And then a few years ago we had more deriding of the NCTM’s new new math with a renewed focus on teaching more conceptually than procedurally, especally by the creators of the website Mathematically Correct who were parents whose careers involve regular use of mathematics, among others. Most people seem to hate the development of the left side and the problem is ultimately one of motivation rather than content. (Sorry, Robert, if I am a little off topic, but I keep reading your tweets…)

I teach a three quarter calculus sequence something like this, there are two fifty minute lectures a week and one recitation session of ninety minutes in which no new material is presented. Students can also go to one optional tutorial session conducted by a TA.

I’m not a teacher but I’ve thought about this problem quite a lot, mostly due to talking to non-math people who are horrified by even the most basic calculus. Most people think basic math is much more difficult than it actually is, which doesn’t help(“interest is aptitude” as they say). Mathematical types love to sadistically dive in right away with a formal proof approach, basing the entire course on that tone. And they wonder why the majority of non-math types absolutely dread mathematics. Which is tragic, as I think knowing at least the basics of calculus is one of the most valuable, and even beautiful things worth knowing.

So here’s my crazy approach, and I have no idea if it would work in practice.
Tell the story of calculus, according to its historical progression. People can naturally remember stories anchored to real people, and real events. How many courses after calculus does it take to finally get the context to fully appreciate everything that is introduced in freshman calculus? A lot. Since this context can’t be expected to be given in 100 hours, stories are the next best means for giving people some kind of contextual framework in which they can relate and remember the core concepts of calculus.

There have been a few books I’ve recently seen which present calculus as it was historically developed(these are not beginners books, maybe senior or grad level). The books were: Mathematics and its History by Stillwell, and Analysis by its History by Hairer and Wanner. For me the single most interesting aspect presented in these books is that modern calculus is taught historically backwards. The historical progression was Series(Viete, Gregory) -> Integration(Fermat) -> Differentiation(Newton, Liebniz) -> Limits(Cauchy). Another consequence I’ve seen from the backwards approach is that all the best example problems which these mathematical masters struggled with in developing calculus are discarded, or modified so as to be unrecognizable. It’s maybe a handful of “showcase” problems, but in solving them it illuminates the kind of reasoning which established calculus step by step. And these are not “tough” problems by todays standards–the benefit of hindsight really helps. It’s much easier to remember 10-15 core problems which provide this initial insight than to go through the rote process of solving loosely connected problem sets of about 30 problems.

Perhaps the #1 retort people give when asking them to learn math is “how will I ever use this?” But if they don’t really know what it’s used for, how can they know they won’t need to use it someday? I think it’s much more insightful to be told “here, this is a problem which the nobility in France struggled with, and here is how Fermat solved it, and it ended up being useful and later gave Newton such and such an idea.”

Also, every single calculus course I’ve taken absolutely fell apart in methods of integration(I moved a lot, switched schools a few times, had to re-take freshmen calculus). There is no reason to even mention hyperbolic functions, and probably little reason to present the full machinery of trig integrals. (Preparation for Jacobi’s invariant & ellipitical integrals is way off target). Integration by parts is probably the only one that matters because
it has so many uses in series expansions. And skip parameterizing functions to prepare for the vector calculus machinery(which only even makes sense once you get to differential forms and see the real generalized Stokes theorem where independence of parameter is the entire point). It’s too bad that this 2nd half of introductory calculus is made so awful in standard textbooks–whatever students understand by then is quickly undone by trying to present integration as if it were a single, coherent topic. (Anyone who reaches Lebesgue, measure theory and calculus of variations knows it’s not).

Whew! This got much longer than I expected, and it’s pretty general, but I hope it’s helpful, even to others reading this blog.